Representations of braid group obtained from rational conformal field theories can be used to obtain explicit representations of Temperley-Lieb-Jones algebras. The method is described in detail for SU(2) k Wess-Zumino conformal field theories and its generalization to an arbitrary rational conformal field theory outlined. Explicit definition of an associated linear trace operation in terms of a...

We study four-point correlation functions with logarithmic behaviour in Liouville field theory on a sphere, which consist of one kind of the local operators. We study them as non-integrated correlation functions of the gravitational sector of two-dimensional quantum gravity coupled to an ordinary conformal field theory in the conformal gauge. We also examine, in the (p, q) minimal conformal fie...

In a short review of recent work, we discuss the general problem of constructing the actions of new conformal field theories from old conformal field theories. Such a construction follows when the old conformal field theory admits new conformal stress tensors in its chiral algebra, and it turns out that the new conformal field theory is generically a new spin-two gauge theory. As an example we ...

In these mostly expository lectures, we give an elementary introduction to conformal field theory in the context of probability theory and complex analysis. We consider statistical fields, and define Ward functionals in terms of their Lie derivatives. Based on this approach, we explain some equations of conformal field theory and outline their relation to SLE theory.

The ADE graphs – the root diagrams of the simply laced simple Lie algebras appeared in the framework of the two-dimensional conformal field theories (CFT) based on the affine algebra sl(2) k in the classification of modular invariant partition functions [1] and in some related lattice models [2]. The modular invariants are labelled by the Coxeter number h = k + 2 while their diagonal terms, enc...

Let’s begin by recalling a fact about 2d conformal field theories from Phil’s talk last week. Firstly, we can define a complex semigroup A whose objects are conformal equivalence classes of Riemann surfaces which are diffeomorphic to annuli, equipped with parameterisations of their boundary components, one incoming and one outgoing. The semigroup operation is given by gluing an outgoing to an i...